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Diffstat (limited to 'info/mathtrip/src/series3.tex')
-rw-r--r-- | info/mathtrip/src/series3.tex | 79 |
1 files changed, 79 insertions, 0 deletions
diff --git a/info/mathtrip/src/series3.tex b/info/mathtrip/src/series3.tex new file mode 100644 index 0000000000..b9a10034c0 --- /dev/null +++ b/info/mathtrip/src/series3.tex @@ -0,0 +1,79 @@ +%The following command typeset the series of the second +%column of the page 9 +% +%This command has one parameter: +% 1) The width of the mathematical text +\newcommand\TNineSeriesThree[1]{% + %This command typeset a fact about a serie and + %the math used to describe the property. + % + %The command has two parameters: + % 1) The title of the fact + % 2) The corresponding math. + %This macro uses the global (to the macro) variables '\Hspace', '\CurrentLineWidth' + %which should contain the space used to typeset the math. + \def\SerieProperty##1##2{% + \def\temp{\TNineTitle{##1} \ensuremath{##2}}% + \savebox\TmpBoxA{\temp}% + \settowidth\CurrentLineWidth{\usebox\TmpBoxA}% + \ifdimcomp{\CurrentLineWidth}{<}{\HSpace}% + {%Enough space, everything is written on one line + \temp + }{%The title is on one line, the math typeset as 'display math' + \TNineTitle{##1}% + \begin{displaymath}% + ##2% + \end{displaymath}% + }% + \par + \AdjustSpace{\TNineInterTitle}% + }% + %This command typeset a line showing the expansion of ordinary power serie + % + %The command has 2 parameters + % 1) The formula to be expanded + % 2) The expanded version + \def\LineOfOrdPowSerie##1##2{##1&=&##2\\[\TNineExpansionSkip]}% + \parbox[t]{#1}{% + \TNineSeriesFontSize + \DisplaySpace{\TNineDisplaySpace}{\TNineDisplayShortSpace}% + \setlength{\HSpace}{#1}% + \SerieProperty{Ordinary power series:} + {A(x) = \sum_{i=0}^\infty a_i x^i} + \SerieProperty{Exponential power series:} + {A(x) = \sum_{i=0}^\infty a_i \frac{x^i}{i!}} + \SerieProperty{Dirichlet power series:} + {A(x) = \sum_{i=1}^\infty \frac{a_i}{i^x}} + \SerieProperty{Binomial theorem:} + {(x + y)^n = \sum_{k=0}^n\binom{n}{k} x^{n-k} y^k} + \SerieProperty{Difference of like powers:} + {x^n - y^n = (x-y) \sum_{k=0}^{n-1} x^{n-1-k} y^k} + \SerieProperty{For ordinary power series:} + {\begin{array}{lcl} + \LineOfOrdPowSerie{\alpha A(x) + \beta B(x)}% + {\sum_{i=0}^\infty(\alpha a_i + \beta b_i) x^i} + \LineOfOrdPowSerie{x^k A(x)}% + {\sum_{i=k}^\infty a_{i-k} x^i} + \LineOfOrdPowSerie{\frac{A(x) - \sum_{i=0}^{k-1} a_i x^i}{x^k}}% + {\sum_{i=0}^\infty a_{i+k} x^i} + \LineOfOrdPowSerie{A(cx)}% + {\sum_{i=0}^\infty c^i a_i x^i} + \LineOfOrdPowSerie{A'(x)}% + {\sum_{i=0}^\infty (i+1) a_{i+1} x^i} + \LineOfOrdPowSerie{x A'(x)}% + {\sum_{i=1}^\infty i a_i x^i} + \LineOfOrdPowSerie{\int A(x) \, dx}% + {\sum_{i=1}^\infty \frac{a_{i-1}}{i} x^i} + \LineOfOrdPowSerie{\frac{A(x) + A(-x)}{2}}% + {\sum_{i=0}^\infty a_{2i} x^{2i}} + \LineOfOrdPowSerie{\frac{A(x) - A(-x)}{2}}% + {\sum_{i=0}^\infty a_{2i+1} x^{2i+1}} + \end{array} + } + \SerieProperty{Summation:} + {\text{If }b_i = \sum_{j=0}^i a_i\text{ then } + B(x) = \frac{1}{1 -x} A(x)} + \SerieProperty{Convolution:} + {A(x)B(x) = \sum_{i=0}^\infty\left(\sum_{j=0}^i a_j b_{i-j}\right)x^i} + } +} |